* It's much farther away. Alpha Centauri is about 4.37 light-years away, which is incredibly far. The Sun is only 8.3 light-minutes away. Distance plays a huge role in how large something appears to us.
* It's actually smaller than the Sun. While Alpha Centauri is a triple star system, its brightest component, Alpha Centauri A, is slightly smaller than our Sun.
To illustrate this: Imagine two coins, one small and one large. Place the small coin close to your face and the large coin far away. While the large coin is physically bigger, the smaller coin will appear much larger because it's closer. The same principle applies to stars.
Even though Alpha Centauri is the closest star system to us, it's still incredibly far away. This makes it appear much smaller than the Sun, even though it's not much smaller in physical size.